On the analytic integrability of the Liénard analytic differential systems

Abstract

We consider the Lienard analytic differential systems $\dot x = y$, $\dot y =-g(x) -f(x)y$, where $f,g: \mathbb{R}\to \mathbb{R}$ are analytic functions and the origin is an isolated singular point. Then for such systems we characterize the existence of local analytic first integrals in a neighborhood of the origin and the existence of global analytic first integrals.